83大學聯考數學（社會組）單選第2題詳解

1. 因為 $\overline{CD}=2\overline{BD}$，可得 $\overline{CD}:\overline{BD}=2:1$。 由分點公式： $$\overset{\large\rightharpoonup}{AD} = \dfrac{1}{3}\overset{\large\rightharpoonup}{AC} + \dfrac{2}{3}\overset{\large\rightharpoonup}{AB}$$ 2. 又 $E$ 為 $\overline{AC}$ 的中點，故 $\overset{\large\rightharpoonup}{AE} = \dfrac{1}{2}\overset{\large\rightharpoonup}{AC}$。 利用向量加法： $$\overset{\large\rightharpoonup}{AD} = \overset{\large\rightharpoonup}{AE} + \overset{\large\rightharpoonup}{ED}$$ $$\dfrac{1}{2}\overset{\large\rightharpoonup}{AC} + \overset{\large\rightharpoonup}{ED} = \dfrac{1}{3}\overset{\large\rightharpoonup}{AC} + \dfrac{2}{3}\overset{\large\rightharpoonup}{AB}$$ $$\overset{\large\rightharpoonup}{ED} = \dfrac{2}{3}\overset{\large\rightharpoonup}{AB} - \dfrac{1}{6}\overset{\large\rightharpoonup}{AC}$$ 由此對照可得 $r = \dfrac{2}{3}$ 且 $s = -\dfrac{1}{6}$， 所以 $$r+s = \dfrac{2}{3} - \dfrac{1}{6} = \dfrac{1}{2}.$$ 答案為選項 $(1)$。